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Astron. Astrophys. 327, 1023-1038 (1997)

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6. What could be the reason for the flat temperature profile ?

The last section showed, that the temperature profile in all nights is flatter than expected from standard outburst models. In the following some possibilities for changing the steepness of the profile are discussed.

6.1. A hole in the disk ?

A possible explanation for the flat temperature profiles could be a hole in the inner disk (Wood et al. 1992). The reconstruction algorithm will nevertheless distribute light inside the region of the hole, thereby lowering intensity outside. In Fig. 7 temperature profiles of the first night (19.7.90) are shown as an example, whereby for the reconstruction no hole and holes with radii of 0.1 [FORMULA], 0.2 [FORMULA] and 0.3 [FORMULA] respectively were assumed. At least a hole of 0.2 [FORMULA] is required to produce clear stationary profiles. The two solid lines in Fig. 7 correspond to theoretical stationary disk temperature profiles, with corresponding mass transfer rates of [FORMULA] and [FORMULA] per year. In Fig. 8 three-dimensional pictures in colour V are visible. The inner Lagrangian point is centered in the middle of the lower right edge of the 3-D grids in each subpicture.

[FIGURE] Fig. 7. Brightness temperature profiles of IP Peg in the colour V for no hole and holes with radii of 0.1 [FORMULA], 0.2 [FORMULA] and 0.3 [FORMULA]
[FIGURE] Fig. 8. 3 - D plots of IP Peg in the V filter for no, 0.1, 0.2 and a 0.3 [FORMULA] hole. (For clarity the ribbon is not plotted here)

Let us now assume that we are looking from the secondary towards the white dwarf. Especially in the 0.3 - hole picture we can see that in the reconstructions light has accummulated in front and behind the white dwarf. That means, that this light is really needed inside the hole. If we make an artificial disk with a hole inside and reconstruct this disk, assuming no hole, we would see the highest intensity in the reconstruction left and right from the white dwarf. This behavior of the eclipse mapping algorithm can clearly be confirmed by simulations.

If there would be a hole in the accretion disk of IP Peg, we would see high intensity left and right of the white dwarf in those reconstructions that assume no hole. But we see the contrary. We see light in front and behind of the white dwarf in reconstructions where a hole is assumed.

The resolution of the eclipse-mapping is limited by the time resolution and the signal-to-noise ratio of the light curves. Reconstructions for all nights show the same behavior. The accumulation of light at certain places around the hole can only be seen for a hole greater than 0.1 [FORMULA]. But all reconstructions assuming no hole are without significant features. That leads to the conclusion, that any hole inside the disk of IP Peg should be smaller than 0.1 [FORMULA]. But such a hole is not big enough to explain the flatness of the temperature profile. Therefore we have to assume that other mechanisms (e.g. a flared disk or material above the disk) lead to a flat temperature profile in the reconstructions or that the disk is really flat.

6.2. A high disk rim ?

The shadowing of the inner disk regions by a very high disk rim could be another explanation for the flat temperature profile.

Smak (1994) already investigated this problem. He found that, for high inclination angles [FORMULA] partial screening of the disk through a disk rim could be responsible for a flat temperature profile. He used a disk build up with rings of constant brightness, which was completely flat, i.e. with a disk opening angle [FORMULA] of [FORMULA] (see Sect. 2.4 in his article). Because our 3-D code gives us the possibility to investigate this problem more deeply, using a full x-y-grid and an opening angle, his results are not totally comparable to ours.

In order to test the influence of an high disk rim, we created light curves from disks with a stationary [FORMULA] profile, but having very high disk opening angles. To get a shadow onto the disk from the outer rim a total disk opening angle [FORMULA] greater than [FORMULA] is necessary (i: inclination-angle of the system, [FORMULA]). This implies that for IP Peg the first shadows from the rim onto the disk occur at a total opening angle [FORMULA] (i.e. [FORMULA] up and down). For the reconstruction a total opening angle [FORMULA] of [FORMULA] (i.e [FORMULA] up and [FORMULA] down) was assumed. Simulations show that the algorithm detects the true temperature distribution for all total opening angles up to [FORMULA] (i.e. [FORMULA] up and down) like shown in Fig. 9A. If the opening angle gets larger, the log-T/log-R profile becomes flat and the reconstructed temperature in the inner disk regions which are permanently obscured is decreasing. But also the amount of light deposited in the disk ribbon is increased dramatically. The log-T/log-R profile for an opening angle [FORMULA] of [FORMULA] (i.e. [FORMULA] up and down) looks similar to the IP Peg reconstructions, but the light curve looks too "round" (see Fig. 9B).

[FIGURE] Fig. 9a-d. A Plots of the reconstruction of a disk with IP Peg parameters and a total disk opening angle of [FORMULA]. The profile of the original disk is identical with the solid line in the log-T/log-R plots in the right panel. This plot shows that the temperature profile is reconstructed very well, even if the real opening angle is [FORMULA] and for the reconstruction an angle of [FORMULA] is assumed. With a opening angle [FORMULA] the first faint shadows from the rim would occure on the disk's surface. B Plots of the reconstruction of a disk with a total disk opening angle of [FORMULA]. The profile of the original disk is identical with the solid line in the log-T/log-R plots in the right panel. The disk opening angle is now high enough to obscure perceptible parts of the inner disk. The log-T/log-R plots show now a flattened profile similar to the IP Peg reconstructions C Plots of the reconstruction of a disk with a 8000 K half-sphere centered on the white dwarf with radius [FORMULA]. The log-T/log-R profile is very similar to that one found for IP Peg. D Plots of the reconstruction of a disk with a 8000 K half-sphere centered on the white dwarf with radius [FORMULA]. The sphere is now big enough to be partially uneclipsed by the secondary. This leads to additional uneclipsed light in the light curve. This uneclipsed light is distributed at the outer rim and away from the inner Lagrangian point (see increase in the brightness temperature at larger radii in the log-T/log-R plots).

The high disk rim could in principle explain the flat log-T/log-R profile, but would require an unrealistically high total opening angle [FORMULA], greater than [FORMULA] (for comparison see the calculation of the scale height H in Sect. 4). Therefore H/R = [FORMULA] must be greater than 0.4, or the rim should be higher than about 10 times the scale height H calculated in Sect. 4, or the temperature must be higher by a factor of 100 (i.e. about 800000 K) to get a new scale height [FORMULA] which is high enough to make a sufficient great shadow.

Another problem is that neither a hole in the disk nor a high disk rim is able to mimic the travelling of any structure in the disk as found in the observations (see Fig. 5 or Fig. 6). We therefore conclude that these two explanations are unlikely.

If the front itself causes an increase in the disk's thickness, such a structure could also shade these inner parts of the disk. However [FORMULA] has to be fulfilled. Note that the decrease of the temperature profile through the influence of a shadow affects only regions that are obscured permanently at all orbital phases by the shadow. Otherwise, the regions would be reconstructed well, as our simulations show.

6.3. An optically thick wind sphere ?

The flat temperature profile we observe may be due to some non-radiative cooling of the inner disk region. One possibility is extraction of energy in the form of a wind from the disk surface. The energy normally radiated by the disk surface is carried off as kinetic energy at the wind's terminal velocity [FORMULA].

The details of the disk wind structure are beyond the scope of this paper. We note, however, that expanding streamlines and possibly increasing outflow velocities mean in general that the density drops outward, and the temperature may also drop as a result of adiabatic expansion. As a result, the opacity of the wind drops rapidly with radius. The photosphere of the wind may tend to occur where the temperature drops below about 8000 Kelvin, allowing hydrogen to recombine and causing a dramatic decrease in the bound-free continuum opacity. This scenario may be a plausible explanation for the inner disk having a roughly constant brightness temperature of 8000 Kelvin, as we deduce from the maps we have fitted to the eclipse light curves.

The wind photosphere may occur close to the disk plane, or it may alternatively occur at some distance above the plane. In the latter case our assumption of a flat disk when fitting the eclipse light curves will not be valid.

In order to test this, we made light curves with a constant 8000 Kelvin half-sphere centered around the white dwarf and having the front radius. The front was calculated as described in the previous section and also outside we assumed a constant 4000 K region. In the reconstructions we assumed a flat disk. The results are shown in Fig. 9C-D.

For the first simulation a sphere of radius [FORMULA] was used. The reconstruction shows a huge increase in intensity in the disk ribbon at the positions which undergo an eclipse at the same time as the sphere. The originally axisymetric intensity distribution is now distorted and shifted away from the inner Lagrangian point to a location opposite to the white dwarf. Because pixels on the sphere are above the disk plane, and tipped more directly toward the earth than pixels in the disk plane, they produce a bright region projected toward the far rim of the disk (light-shifting). The entropy constraint tries to suppress this azimuthal structure, and does so by moving this flux into the disk ribbon. Fig. 9C shows the fitted light curve and the reconstructed temperature profile.

For the second simulation a sphere of radius [FORMULA] was used. The reconstruction shows in addition to the above results a half-ring structure in the outer disk rim, opposite to the white dwarf. This effect is due to the uneclipsed light that comes from the sphere. For a system with IP Peg parameters a sphere with radius [FORMULA] is too big to be eclipsed totally at phase zero. Fig. 9D shows the fitted light curve and the reconstructed temperature profile (see figure captions for details).

The IP Peg reconstructions do not show any signs of additional uneclipsed light or a light-shifting during the outburst. On the other hand an opaque wind photosphere could shade the inner disk regions and if related to the regions inside the cooling front mimic the IP Peg results. But then some geometrical constraints have to be fulfilled. The wind-sphere should more be a wind-layer than a sphere. It should be flat enough at every state of the (observed) outburst to be totally eclipsed, and its surface elements should not be tilted to much towards the observer to avoid light-shifting in the reconstructions.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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